Almost sure existence of global weak solutions for incompressible MHD equations in negative-order Sobolev space

In this paper we consider the Cauchy problem of the incompressible Magnetohydrodynamics (MHD) equations in the periodic space domain TN, where N≥2. After a suitable randomization to the initial data, we obtain the almost sure existence of global weak solutions with the initial data in Hs(TN), s∈(−1,0). Specially, the global weak solution is unique when N=2.